지겔 모듈라 형식

수학노트
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지겔 상반 공간

  • 지겔 상반 공간 \(\mathcal{H}_g\)

\[ \mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\mathrm{T}}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\} \]

  • 사교군 \(\Gamma_g:={\rm Sp}(2g,\Z)\)
  • \(\mathcal{A}_g=\mathcal{H}_g/\Gamma_g\) : moduli space of principally polarized abelian varieties


지겔 모듈라 형식

정의

weight이 k이고 genus(또는 degree)가 \(g\)인 지겔 모듈라 형식은 다음 조건을 만족하는 해석함수 \(f:\mathcal{H}_g\to \mathbb{C}\)로 정의된다 \[ f \left( (A\tau +B)(C\tau + D)^{-1}\right) = \det(C\tau +D)^{k} f(\tau),\, \forall \begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in {\rm Sp}(2g,\Z) \]

푸리에 전개

  • 지겔 모듈라 형식 \(f\in M_k(\Gamma_g)\)는 다음과 같은 형태의 푸리에 전개를 가진다

\[f(\tau)=\sum_{T}a(T)\exp\left(2\pi i \operatorname{Tr}(T\tau)\right)\] 여기서 \(T\in \operatorname{Mat}_2(\frac{1}{2}\mathbb{Z})\)는 대각성분이 정수인 대칭행렬.

Kocher 원리

지겔 모듈라 형식 \(f\in M_k(\Gamma_g)\)의 푸리에 전개에서, \(T\)가 positive semi-definite 행렬이 아니면, \(a(T)=0\)이다


지겔 모듈라 형식의 예


관련된 항목들


매스매티카 파일 및 계산 리소스

사전 형태의 자료


리뷰, 에세이, 강의노트


관련도서

  • Andrianov, Anatoli. Introduction to Siegel Modular Forms and Dirichlet Series Springer, 2010.
  • Klingen, Helmut. Introductory Lectures on Siegel Modular Forms. Cambridge University Press, 1990.
  • Maass, Lectures on Siegel's Modular Functions


관련논문

  • Andrew Knightly, Charles Li, On the distribution of Satake parameters for Siegel modular forms, arXiv:1605.03792 [math.NT], May 12 2016, http://arxiv.org/abs/1605.03792
  • Satoshi Wakatsuki, The dimensions of spaces of Siegel cusp forms of general degree, arXiv:1602.05676 [math.NT], February 18 2016, http://arxiv.org/abs/1602.05676
  • Gerard van der Geer, Exploring modular forms and the cohomology of local systems on moduli spaces by counting points, arXiv:1604.02654 [math.AG], April 10 2016, http://arxiv.org/abs/1604.02654
  • Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi, An equidistribution theorem for holomorphic Siegel modular forms for \(GSp_4\), arXiv:1604.02036[math.NT], April 07 2016, http://arxiv.org/abs/1604.02036v1
  • Dickson, Martin J. “Hecke Eigenvalues of Klingen--Eisenstein Series of Squarefree Level.” arXiv:1512.09069 [math], December 30, 2015. http://arxiv.org/abs/1512.09069.
  • Ichikawa, Takashi. “Integrality of Nearly (holomorphic) Siegel Modular Forms.” arXiv:1508.03138 [math], August 13, 2015. http://arxiv.org/abs/1508.03138.
  • Schulze-Pillot, Rainer. “Averages of Fourier Coefficients of Siegel Modular Forms and Representation of Binary Quadratic Forms by Quadratic Forms in Four Variables.” arXiv:1202.4909 [math], February 22, 2012. http://arxiv.org/abs/1202.4909.
  • Piazza, Francesco Dalla, Alessio Fiorentino, Samuel Grushevsky, Sara Perna, and Riccardo Salvati Manni. ‘Vector-Valued Modular Forms and the Gauss Map’. arXiv:1505.06370 [math], 23 May 2015. http://arxiv.org/abs/1505.06370.
  • Heim, Bernhard, and Atsushi Murase. "On the Igusa modular form of weight 10." 数理解析研究所講究録 1767 (2011): 179-187. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1767-18.pdf
  • Tsuyumine, Shigeaki. “Thetanullwerte on a Moduli Space of Curves and Hyperelliptic Loci.” Mathematische Zeitschrift 207, no. 1 (May 1, 1991): 539–68. doi:10.1007/BF02571407.
  • Tsuyumine, Shigeaki. 1986. “On Siegel Modular Forms of Degree Three.” American Journal of Mathematics 108 (4): 755. doi:10.2307/2374517.
  • Igusa, Jun-Ichi. 1962. “On Siegel Modular Forms of Genus Two.” American Journal of Mathematics 84 (1): 175. doi:10.2307/2372812.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'siegel'}, {'LOWER': 'modular'}, {'LEMMA': 'form'}]
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